Abstract: Let X be a completely regular Hausdorff space, let L be the linear space of all finite linear combinations of the point measures on X and let denote the space of Baire measures on X. The following is proved: Ifis endowed with the topology of uniform convergence on the uniformly bounded, equicontinuous subsets of, thenis a complete locally convex space in which L is dense and whose dual is, provided there are no measurable cardinals. A complete description of the situation in the presence of measurable cardinals is also given. Let be the subspace of consisting of those measures which have compact support in the realcompactification of X. The following result is proved: Ifis endowed with the topology of uniform convergence on the pointwise bounded and equicontinuous subsets of, thenis a complete locally convex space in which L is dense and whose dual is, provided there are no measurable cardinals. Again the situation if measurable cardinals exist is described completely. Let M denote the Banach dual of . The following is proved: If M is endowed with the topology of uniform convergence on the norm compact subsets of, then M is a complete locally convex space in chich L is dense. It is also proved that is metrizable if and only if X is discrete and that the metrizability of either or M is equivalent to X being finite. Finally the following is proved: Ifhas the Mackey topology for the pair, thenis complete and L is dense in.